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On the Geometry of Projective Tensor Products ON THE GEOMETRY OF PROJECTIVE TENSOR PRODUCTS OHAD GILADI, JOSCHA PROCHNO, CARSTEN SCHUTT,¨ NICOLE TOMCZAK-JAEGERMANN, AND ELISABETH WERNER n Abstract. In this work, we study the volume ratio of the projective tensor products `p ⊗π n n `q ⊗π `r with 1 ≤ p ≤ q ≤ r ≤ 1. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kaˇsintype whenever 1 ≤ p ≤ q ≤ r ≤ 2 or 1 ≤ p ≤ 2 ≤ r ≤ 1 and q = 2. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products `n ⊗ · · · ⊗ `n with k 2 p1 π π pk N and 1 ≤ p1 ≤ · · · ≤ pk ≤ 1. 1. Introduction In the geometry of Banach spaces the volume ratio vr(X) of an n-dimensional normed space X is defined as the n-th root of the volume of the unit ball in X divided by the volume of its John ellipsoid. This notion plays an important role in the local theory of Banach spaces and has significant applications in approximation theory. It formally originates in the works [Sza78] and [STJ80], which were influenced by the famous paper of B. Kaˇsin[Kaˇs77] on nearly Euclidean orthogonal decompositions. Kaˇsindiscovered that for arbitrary n 2 N, 2n the space `1 contains two orthogonal subspaces which are nearly Euclidean, meaning that n their Banach-Mazur distance to `2 is bounded by an absolute constant. S. Szarek [Sza78] n noticed that the proof of this result depends solely on the fact that `1 has a bounded volume n ratio with respect to `2 . In fact, it is essentially contained in the work of Szarek that if X is a 2n-dimensional Banach space, then there exist two n-dimensional subspaces each n having a Banach-Mazur distance to `2 bounded by a constant times the volume ratio of X squared. This observation by S. Szarek and N. Tomczak-Jaegermann was further investigated in [STJ80], where the concept of volume ratio was formally introduced, its connection to the cotype 2 constant of Banach spaces was studied, and Kaˇsintype decompositions were proved n n for some classes of Banach spaces, such as the projective tensor product spaces `p ⊗π `2 , 1 ≤ p ≤ 2. Given two vector spaces X and Y , their algebraic tensor product X ⊗ Y is the subspace of the dual space of all bilinear maps on X × Y spanned by elementary tensors x ⊗ y, x 2 X, y 2 Y (a formal definition is provided below). The theory of tensor products was established by A. Grothendieck in 1953 in his R´esum´e[Gro53] and has a huge impact on Date: April 6, 2017. 2010 Mathematics Subject Classification. 46A32, 46B28. O.G. was supported in part by the Australian Research Council. J.P. was supported in part by the FWF grant FWFM 162800. A part of this work was done when N.T-J. held the Canada Research Chair in Geometric Analysis; also supported by NSERC Discovery Grant. E.W. was supported by NSF grant DMS-1504701. 1 Banach space theory (see, e.g., the survey paper [Pis12]). This impact and the success of the concept of tensor products is to a large extent due to the work [LP68] of J. Lindenstrauss and A. Pe lczynskiin the late sixties who reformulated Grothendieck's ideas in the context of operator ideals and made this theory accessible to a broader audience. Today, tensor products appear naturally in numerous applications, among others, in the entanglement of qubits in quantum computing, in quantum information theory in terms of (random) quantum channels (e.g., [AS06,ASW10,ASW11,SWZ11]) or in theoretical computer science to represent locally decodable codes [Efr09]. For an interesting and recently discovered connection between the latter and the geometry of Banach spaces we refer the reader to [BNR12]. The geometry of tensor products of Banach spaces is complicated, even if the spaces involved are of simple geometric structure. For example, the 2-fold projective tensor product of Hilbert spaces, `2 ⊗π `2, is naturally identified with the Schatten trace classpS1, the space ∗ of all compact operators T : `2 ! `2 equipped with the norm kT kS1 = trace T T . This space does not have local unconditional structure [GL74]. The geometric structure of triple tensor products is even more complicated and therefore it is hardly surprising that very little is known about the geometric properties of these spaces. For instance, regarding the permanence of cotype (see below for the definition) under projective tensor products, it was proved by N. Tomczak-Jaegermann in [TJ74] that the space `2 ⊗π `2 has cotype 2, but the corresponding question in the 3-fold case is still open for more than 40 years. G. Pisier proved in [Pis90] and [Pis92] that the space Lp ⊗π Lq has cotype max (p; q) if p; q 2 [2; 1) and, till the present day, it is unknown whether these spaces have a non-trivial cotype when p 2 (1; 2) and q 2 (1; 2]. In the recent paper [BNR12] by J. Bri¨et,A. Naor and O. Regev 1 1 1 they showed that the spaces `p ⊗π `q ⊗π `r fail to have non-trivial cotype if p + q + r ≤ 1. Their proof uses deep results from the theory of locally decodable codes. A direct, rather surprising consequence of their work is that for p 2 (1; 1) the space `p ⊗π `2p=(p−1) ⊗π `2p=(p−1) fails to have non-trivial cotype, while, by Pisier's result, the 2-fold projective tensor product `2p=(p−1) ⊗π `2p=(p−1) has finite cotype. Let us also mention that interest in cotype is, to a great deal, due to a famous result of B. Maurey and G. Pisier [MP76], who showed that a n Banach space X fails to have finite cotype if and only if it contains `1's uniformly. In view of the various open questions and surprising results around the geometry of projec- tive tensor products, with the present paper, we contribute to a better understanding of the n n n geometric structure of 3-fold projective tensor products `p ⊗π `q ⊗π `r (1 ≤ p ≤ q ≤ r ≤ 1) by studying one of the key notions in local Banach space geometry, the volume ratio of these spaces. This provides new structural insight and allows, on the one hand, to draw conclusions regarding cotype properties of these spaces and, on the other hand, to see which of these spaces allow a nearly Euclidean decomposition of Kaˇsintype. 2. Presentation of the main result Given two Banach spaces, (X; k · kX ) and (Y; k · kY ), the projective tensor product space, denoted X ⊗π Y , is the space X ⊗ Y equipped with the norm ( m m ) X X kAkX⊗πY = inf kxikX kyikY : A = xi ⊗ yi; xi 2 X; yi 2 Y : (2.1) i=1 i=1 See Section 3.3 below for more information on projective tensor products. 2 Given an n-dimensional Banach space (X; k · kX ), let BX denote its unit ball, and let EX be the ellipsoid of maximal volume contained in BX . The volume ratio of X is defined by vol (B )1=n vr(X) = n X ; (2.2) voln(EX ) where voln(·) denotes the n-dimensional Lebesgue measure. Considering the importance of tensor products, it is natural to study the geometric prop- erties of X ⊗π Y and, in particular, the volume ratio vr(X ⊗π Y ) of these spaces. As already n n mentioned in the introduction, when X = `p (1 ≤ p ≤ 2) and Y = `2 this was carried out in [STJ80]. The complete answer was given later by C. Sch¨uttin [Sch82], where it was proved that if 1 ≤ p ≤ q ≤ 1, then 8 >1; q ≤ 2; > 1 − 1 >n 2 q ; p ≤ 2 ≤ q; 1 + 1 ≥ 1; n n < p q vr `p ⊗π `q p;q 1 − 1 n p 2 ; p ≤ 2 ≤ q; 1 + 1 ≤ 1; > p q > max 1 − 1 − 1 ;0 :>n ( 2 p q ); p ≥ 2: The notation p;q means equivalence up to constants that depend only on p and q. In [DM05] this was generalized by A. Defant and C. Michels to the setting E ⊗π F , where E and F are symmetric Banach sequence spaces, each either 2-convex or 2-concave. n n n We study tensor products `p ⊗π `q ⊗π `r (1 ≤ p ≤ q ≤ r ≤ 1). Our main result is as follows. Theorem A. Let n 2 N and 1 ≤ p ≤ q ≤ r ≤ 1. Then we have 8 >1; r ≤ 2; > 1 1 1 > max( 2 − q − r ;0) 1 1 1 n n n <n ; p ≤ 2 ≤ q; p + q + r ≥ 1; vr `p ⊗π `q ⊗π `r p;q;r 1 − 1 n p 2 ; p ≤ 2 ≤ q; 1 + 1 + 1 ≤ 1; > p q r > max 1 − 1 − 1 − 1 ;0 :n ( 2 p q r ); p ≥ 2: In the case p ≤ q ≤ 2 ≤ r, we have 1 1 1 1 n n n min( 2 − r ; q − 2 ) 1 .p;q;r vr `p ⊗π `q ⊗π `r .p;q;r n : Here and in what follows .p;q;r, &p;q;r mean inequalities with implied positive constants that depend only on the parameters p; q; r. The asymptotic notation p;q;r means that we have both .p;q;r and &p;q;r.
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